User blog:Hyp cos/Googolisms hard to extend?
Here, to extend a googolism (large number) means to make a larger number in a similar way. Some googolisms from "notations" such as array notations, hydras, FGH/HH/SGH + OCF, are easy to extend. For instance, Bird's \(U^{U(3)}(3)\) can extend to a new part of Bird's array notation, with arrays as subscript of the slash. Extending to large branches of new numbers is not much more difficult than understanding the notation itself. Some googolisms resulting from "combinatorial functions" are also easy to extend, though naive extension. For instance, from TREE(3) there are simply TREE(4), TREE(5), etc. But there are still some numbers appearing singly, or even without any "input parameters", thus hard to extend. A cool example is 808017424794512875886459904961710757005754368000000000, which can be defined as "the largest order of finite simple groups that are not cyclic, alternative, or of 16 families of group of Lie type". Are there another googolism of this kind, i.e. hard to extend? My collection of those numbers The following list contains numbers with mathematical properties unrelated to the use of base 10, and hard to extend. *The logarithmic density of such positive integers x that \(\pi(x)>\text{li}(x)\), approximately 0.00000026 *210, largest triangular and pentatopic number *454, largest integer not to be sum of 7 nonnegative cubes *952, largest integer not to be sum of 27 nonnegative fifth powers *4095, largest triangular and Mersenne number *4900, largest square and pyramidal number *5778, largest triangular and Lucas number *7140, largest trianglar and tetrahedral number *8191, largest more-than-two-digit repunit prime in more than one base *11628, largest triangular and 5-simplex number *13792, largest integer not to be sum of 16 fourth powers *19600, largest square and tetrahedral number *24310, largest triangular and 8-simplex number *196883, minimal number of dimensions of a crystal lattice whose symmetry rotations and reflections form the Monster group *208335, largest triangular and pyramidal number *9653449, largest square and stella octangula number *34283340, largest triangular number \(T_{x^2-1}\) such that \(T_{x^2-1}/6\) is also triangular *195643523275200, largest highly composite number whose amount of divisors is also highly composite *808017424794512875886459904961710757005754368000000000, order of Monster group, the largest finite simple group that is not cyclic, alternative, or Lie type The following list contains numbers with base-10-specific properties, and hard to extend. *3435, largest number equal to the sum of each digit raised to its power *40585, largest number equal to the sum of factorial of its digits *23456789, largest prime with consecutive increasing digits *73939133, largest number of whose digits every initial segment forms a prime *389645271, largest "flexible power selfie number" (389645271 = 99+87+76+62+55+44+38+21+13) *3816547290, the only polydivisible number containing each of digits 0~9 *9814072356, largest perfect power without a digit occurring more than once *12157692622039623539, largest number equal to the sum of consecutive powers of its digits *3608528850368400786036725, largest polydivisible number, meaning that any n leading digits form a multiple of n *1033, largest power of 10 as the product of two numbers with no zero digit *115132219018763992565095597973971522401, largest Armstrong number, meaning it equal to the sum of the (amount of its digits)th power of its digits The following list contains physical constants. *Reciprocal fine-structure constant, approximately 137.035999138 *Mass ratio between a proton and an electron, approximately 1836.15267 *Reciprocal gravitational coupling constant based on a proton, approximately 1.693206×1038 *Ratio between the strength of gravitational and electric attraction of a proton and an electron, approximately 2.26873×1039 *Number of atoms in the observable universe, approximately 1080 *Number of photons in the observable universe, approximately 1.1×1089 *Number of fundamental particles in the observable universe, approximately 1097 *Number of subatomic particles needed to fill all space of the universe, approximately 10110 *Volume of the observable universe in Planck units, approximately 8.72×10184 *Space-time volume of the observable universe in Planck units, approximately 1.75×10245 *The single-perturbation count, approximately 1.41×10408 *Number of "vacua" in the string theory landscape within eternal chaotic inflation models, approximately 10500 *Size of a universe giving rise to spontaneous life, approximately 1022650000000 *Estimated size of inflationary universe, approximately 101012 *Estimated number of distinguishable parallel universes, approximately 101016 *Estimated number of universes in the multiverse, approximately 101077 *Estimated distance to an indistinguishable copy of the observable universe, approximately 1010115 *Estimated number of distinct universes in the "string theory landscape", approximately 1010375 *Estimated number of "different types of universes" arising from a certain "eternal cosmic inflation" scenario, approximately 1010107 *Poincare recurrence time of a black hole with mass of the observable universe, approximately 10101.7×10120 *Cosmological inflationary size of the universe, approximately 101010122 *Poincare recurrence time of a black hole with mass of a Linde-type super-inflationary universe, approximately 10101.51×103883775501690 *Poincare recurrence time associated with largest estimated cosmological inflationary, approximately 1010101010122 Category:Blog posts